Weak linearization of the lambda calculus

We identify a restricted class of terms of the lambda calculus, here called weak linear, that includes the linear lambda-terms keeping their good properties of strong normalization, non-duplicating reductions and typability in polynomial time. The advantage of this class over the linear lambda-calculus is the possibility of transforming general terms into weak linear terms with the same normal form. We present such transformation and prove its correctness by showing that it preserves normal forms.